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Infinite sequence with exactly 2009 different numbers

Source: Czech-Polish-Slovak Match, 2009

August 21, 2011

inductionalgebra unsolvedalgebra

Problem Statement

For positive integers

a

and

k

, define the sequence

a_1,a_2,\ldots

by

a_1=a,\qquad\text{and}\qquad a_{n+1}=a_n+k\cdot\varrho(a_n)\qquad\text{for } n=1,2,\ldots

where

\varrho(m)

denotes the product of the decimal digits of

m

(for example,

\varrho(413)=12

and

\varrho(308)=0

). Prove that there are positive integers

a

and

k

for which the sequence

a_1,a_2,\ldots

contains exactly

2009

different numbers.

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